Analysis of ambient noise energy distribution and phase...

Geophys. J. Int. (2009) doi: 10.1111/j.1365-246X.2009.04329.x

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Analysis of ambient noise energy distribution and phase velocity biasin ambient noise tomography, with application to SE Tibet

Huajian Yao and Robert D. van der HilstDepartment of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, MA, USA. E-mail: [email protected]

Accepted 2009 July 7. Received 2009 July 3; in original form 2009 February 13

S U M M A R YGreen’s functions (GFs) of surface wave propagation between two receivers can be estimatedfrom the cross-correlation of ambient noise under the assumption of diffuse wavefields orenergy equipartitioning. Interferometric GF reconstruction is generally incomplete, however,because the distribution of noise sources is neither isotropic nor stationary and the wavefieldsconsidered in the cross-correlation are generally non-diffuse. Furthermore, medium complex-ity can affect the empirical Green’s function (EGF) from the cross-correlation if noise sourcesare all far away (i.e. approximately plane-wave sources), which makes the problem non-linear.We analyse the effect of uneven ambient noise distribution and medium heterogeneity andazimuthal anisotropy on phase velocities measured from EGFs with an asymptotic plane wave(far-field) approximation (which underlies most constructions of phase velocity maps). Phasevelocity bias due to uneven noise distribution can be determined (and corrected) if the noise en-ergy distribution and the velocity model are known. We estimate the (normalized) azimuthaldistribution of ambient noise energy directly from the cross-correlation functions obtainedthrough ambient noise interferometry. The (smaller, second order) bias due to non-linearitycan be reduced iteratively, for instance by using the tomographic model that results from theinversion of uncorrected data. We illustrate our method for noise energy estimation, phasevelocity bias suppression, and ambient noise tomography (including azimuthal anisotropy)with data from a seismic array (26 stations) in SE Tibet. We show that the phase velocity biasdue to uneven noise energy distribution (and medium complexity) in SE Tibet has a smalleffect (

2 H. Yao and R. D. van der Hilst

surface wave GFs). Recent surface wave studies show that the actual distribution of ambient noise energy is, in general, neither isotropic norstationary but reveals directional and temporal variations (Stehly et al. 2006; Yang & Ritzwoller 2008a; Yao et al. 2009). As a consequence,the EGF is generally an incomplete reconstruction of the true GF, which can be manifest, for instance, in a lack of reciprocity of the EGFsbetween two receivers (e.g. Yao et al. 2006).

The uneven distribution of noise sources can produce a bias in the surface wave group or phase velocities measured from the EGFs.Furthermore, if the predominant noise sources are all far away so that wave energy propagating across the receiver array can be approximatedas plane waves, the (unknown) medium heterogeneity can also produce traveltime bias from the EGFs even for an isotropic distribution ofnoise energy (e.g. Tsai 2009). However, in case of spatially homogeneous distribution of noise sources or diffuse wavefields, we would expectaccurate traveltime measurements from the EGFs regardless of any medium complexity. Comparisons of phase velocities from ambientnoise interferometry and traditional two-station analysis (Yao et al. 2008) and of phase velocity maps from ambient noise tomography andteleseismic surface wave tomography (Yang & Ritzwoller 2008b) as well as numerical simulations for certain types of noise distribution(Lin et al. 2008; Yang & Ritzwoller 2008a) indicate that the bias is, indeed, small for the types of study done so far. It appears that theuse of long time windows for cross-correlation (e.g. 1 yr) and temporal and spectral normalization before cross-correlation, for example,one-bit (Shapiro & Campillo 2004) or running-absolute-mean normalization and whitening (Bensen et al. 2007), are efficient in making thedistribution of ambient noise sources more isotropic. Any remaining non-isotropic component of the ambient noise distribution will, however,result in an azimuth-dependence of bias of the phase or group velocity measurements from EGFs, and these discrepancies may be significantfor high-resolution studies or for determination of azimuthal and radial anisotropy.

de Hoop & Solna (2008) present a multiscale analysis of GFs estimated from ‘field–field’ correlations in a random medium. Closer tothe problem of interest here, Tsai (2009) corrects phase or group velocity bias using estimated traveltimes between two stations for givendistribution of (ambient noise) source intensity. With synthetic data he shows that both the azimuthally uneven distribution of source intensityand the heterogeneity of the medium can produce bias in surface wave dispersion measurements. Stehly et al. (2006) and Yang & Ritzwoller(2008a) investigated the azimuth-dependence of ambient noise energy using a back projection method. Brzak et al. (2008) introduced amigration approach to estimate the noise energy distribution from the cross-correlation functions. Harmon et al. (2008) and Yao et al. (2009)used a plane-wave beamforming analysis to estimate the ambient noise energy distribution. However, all these approaches still did not providea quantitative measure of ambient noise energy distribution directly related to the recovery of surface wave GF from the cross-correlation,which is needed to actually correct the bias.

We describe here an approach to estimate the energy distribution of ambient noise and to correct the phase velocity bias for ambientnoise surface wave array tomography. We illustrate the problems—and solutions—with data from a seismograph array in SW China (Yao et al.2006, 2008). The accuracy of GF reconstruction from ambient noise interferometry depends on (i) the azimuthal distribution of noise energy(even versus uneven), (ii) the type of medium (homogeneous versus heterogeneous; isotropic versus anisotropic) in case of plane-wave sourcedistribution and (iii) the scales in the data (frequency) and medium (structural length scales). It is impractical to explore all possible casesbut the examples shown here give insight in the type and magnitude of the problem and the promise of the solution. In Section 2, we analysecross-correlations (and associated EGFs) and present our approach to bias estimation through (asymptotic) plane-wave modelling. Section 3investigates phase velocity bias through stationary phase and Fresnel zone arguments for different types of noise distribution and medium.Section 4 presents a damped least-squares inversion that can be used to estimate the azimuthal variations of ambient noise energy from the(given) correlation functions. In Section 5, we apply the proposed method for noise energy estimation and phase velocity bias correctionto array data in SE Tibet with an iterative approach. Finally, we discuss the importance of analysis of phase velocity bias and noise energyestimation for isotropic and anisotropic ambient noise tomography.

2 P L A N E - WAV E M O D E L L I N G A N D I N T E R F E RO M E T RY

We investigate the surface wave part from ambient noise interferometry and ignore other types of waves generated by the noise sources.If the aperture of an array is small compared to the distance to the main noise sources—many ambient noise tomography studies concernarrays a few hundred kilometres across (e.g. SE Tibet, by Yao et al. 2006, 2008; New Zealand, by Lin et al. 2007; South Korea, by Cho et al.2007)—we can approximate the energy from ambient noise sources as plane waves, for instance, in a theoretical framework by Nakahara(2006) and Tsai (2009). This plane wave approximation is also widely adopted in regional surface wave array analysis (e.g. Yang & Ritzwoller2008b). We represent the noise energy as EP (ω, θ ), with ω the angular frequency and θ the azimuth angle of the incoming plane wave passingacross the array. Local scattering due to heterogeneity of the medium within or close to the array is not considered but their contribution isprobably small compared to that from direct waves (like ocean- or earthquake-generated surface waves) (Yao et al. 2009). Note that EP (ω, θ )represents the total energy of the plane waves with azimuth θ propagating across the array, which may have contributions from many ambientnoise sources. We use ‘plane wave energy’ EP (ω, θ ) and ‘ambient noise energy’ as synonymous throughout this paper. For simplicity weassume straight rays but there is no obstruction for expanding the method to include ray bending or finite frequency sensitivities.

We consider seismograph stations at locations A and B in a 2-D elastic medium (Fig. 1). The incoming plane wave with azimuth angleθ and energy EP (ω, θ ) is assumed to be recorded at both stations with equal amplitude—one-bit cross-correlation (e.g. Yao et al. 2006)or normalized cross-correlation (Bensen et al. 2007) removes the effect of geometrical spreading and attenuation. The phase traveltimedifference δt (or phase delay δφ = ωδt) of the plane wave between the two stations will result in a peak in the cross-correlation function. Thesummation of cross-correlation functions from all plane waves with azimuth angle from 0 to 2π produces the final cross-correlation function,

C© 2009 The Authors, GJIJournal compilation C© 2009 RAS

Analysis of bias in ambient noise tomography 3

Figure 1. Geometry of the plane-wave modelling. The incoming plane wave with azimuth angle θ is shown as the red lines with arrow. Red dashed line showsthe wave front of the plane wave, which is perpendicular to the ray paths. The two stations are located at A and B, shown as the black triangles, with the centralpoint at O and azimuth angle ϕ (from A to B). The green patches are the regions with velocity anomalies. The fast direction with the azimuth angle ψ of theazimuthal anisotropy at point O is shown as the blue bar with arrows at both ends.

CAB(ω, t), between stations A and B

CAB(ω, t) =∫ 2π

0EP (ω, θ ) cos [ω(t − δt)] H (t, δt)dθ, (1)

where t is time and H (t, δt) a time domain taper for the cross-correlation function EP (ω, θ ) cos[ω(t − δt)], which is infinite in time (bluesinusoid trace in Fig. 2a). We use for H (t, δt) a simple cosine taper function with peak centred at δt (dashed trace, Fig. 2a)

H (t, δt) ={

0.5 {1 + cos[2π (t − δt)/T ∗]} t ∈ [δt − T ∗/2, δt + T ∗/2]0 elsewhere

, (2)

where T ∗ is the (time) width of H (t, δt). Here we choose T ∗ = 5T , with T = 2π/ω the period of the sinusoid wave. The integrant in eq. (1)denotes the individual cross-correlation function of the incoming plane wave with azimuth angle θ .

For homogeneous and isotropic medium with phase velocity c, the phase delay of the plane wave (with azimuthal angle θ ) betweenstations A and B (Fig. 1) is

δφ = kAB cos(θ − ϕ) = ωAB cos(θ − ϕ)/

c, (3)

where k = ω/c is the wavenumber, AB is the interstation distance and ϕ is the azimuth angle from station A to station B measured fromnorth.

For a homogeneous but azimuthally anisotropic medium, Rayleigh wave phase velocity c can be approximated by c = c0{1 + Accos 2(θ − ψ)}(Smith & Dahlen 1973), where c0 is the transversely isotropic part of the phase velocity, Ac is the amplitude of the azimuthalanisotropy and ψ is the fast direction of the medium. For general heterogeneous and azimuthally anisotropic medium (Fig. 1), the phase delayδφ between two stations is

δφ =∫ B

B′

ω

c(ω, x)dl −

∫ AA′

ω

c(ω, x)dl, (4)

where x is the spatial coordinate and the integration is along the ray path B′B and A′A as shown in Fig. 1.The time derivative of the ambient noise cross-correlation function CAB(ω, t) yields the EGFs G̃AB(ω, t) and G̃BA(ω, t)

dCAB(ω, t)

dt= −G̃AB(ω, t) + G̃BA(ω,−t)

G̃AB(ω, t) = −dCAB(ω, t)dt

t ≥ 0

G̃BA(ω, t) = −dCAB(ω,−t)dt

t ≥ 0. (5)



Figure 2. (a) Taper function H (t, δt), shown as the red dashed trace, as given by eq. (2). The blue trace show the single sinusoid cross-correlation functioncos[ω(t − δt)] of incoming plane wave and the black trace gives the product of H (t, δt) and cos[ω(t − δt)]. (b) Surface wave window function W (t,AB),shown as the red trace, as given by eq. (7). The black trace shows the theoretical Green’s function given by eq. (6).

Under the assumption of isotropic wavefields or homogeneous distribution of noise sources, theoretical studies (e.g. Weaver & Lobkis2004; Snieder 2004; Roux et al. 2005) have demonstrated that the EGF G̃AB(ω, t) [or G̃BA(ω, t)] is equivalent to the real GF GAB(ω, t) [orGBA(ω, t)], that is the GF that would have been recorded at station at B for a point source at A (or GF recorded at station A for a point sourceat B), except for a frequency-dependent amplitude correction. The phase of EGF is the same as that of GF.

In the far field (loosely, several wavelengths apart) the time-domain windowed GF for the surface wave fundamental mode centred atfrequency ω can be approximated by (e.g. Dahlen & Tromp 1998)

GAB(ω, t) = A cos(k̄ABAB − ωt + π/

4)H (t, tAB), (6)

where A is the amplitude of surface wave GF, k̄AB = 1AB∫ B

A k(ω, x)dl the path average wavenumber between A and B, H (t, tAB) the taperdefined as eq. (2), and tAB = (k̄ABAB + π/4)

/ω the phase traveltime of surface wave fundamental mode at frequency ω. In the context of

phase velocity maps, which is an asymptotic concept, we assume great circle propagation of surface waves between A and B.In order to compare the G̃AB(ω, t) and GAB(ω, t) in the same time window, we define a surface wave window function (black trace,

Fig. 2b)

W (t,AB) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1 t ∈ [tL , tU ] = [AB/vmax, AB

/vmin]

0.5{1 + cos[π (t − tU )

/T ]

}t ∈ (tU , tU + T ]

0.5{1 + cos[π (tL − t)

/T ]

}t ∈ [tL − T, tL )

0 elsewhere

, (7)

where vmin and vmax are the minimum and the maximum group velocities to define the main surface wave window, respectively. W (t,AB)can be frequency-dependent. For typical regional ambient noise surface wave tomography in the period band 10–50 s, vmin ≈ 2 km s–1 andvmax ≈ 5 km s–1. For very short periods (e.g. several seconds), the group velocities that define the surface wave window will depend on thewave speeds at shallow depth (e.g. top few kilometres).

Inside W (t,AB) we measure the phase difference (δφAB) between the EGF, G̃AB(ω, t) and the theoretical GF, GAB(ω, t), atfrequency ω

δφAB(ω) = [G̃AB(ω, t)W (t,AB)] − [GAB(ω, t)W (t,AB)], (8)



where is the operator to obtain the phase angle of a trace at frequency ω (for example, one can take the Fourier transform of the trace andcalculate the phase from the real and imaginary part of spectrum at frequency ω). The traveltime difference between EGF and GF is thengiven by δtAB = δφAB(ω)/ω. If δφAB > 0, the surface wave EGF has a phase shift away from zero time with respect to the theoretical surfacewave GF; in other words, the phase traveltime of the empirical surface wave recovered from noise correlation is larger than that of theoreticalsurface wave. In this case the apparent phase velocity obtained from the EGF is slower than that from the theoretical GF. For δφAB or δtABless than zero, we would expect a higher phase velocity from the EGF than that from the theoretical GF. The relative phase velocity bias (μ)between the EGF and GF can then be expressed as

μ = c̃AB − cABcAB

≈ − δtABtAB

= − δφAB(ω)ωtAB

, (9)

where c̃AB and cAB are the surface wave phase velocities of the EGF and GF, respectively.

3 A Z I M U T H D E P E N D E N T P H A S E V E L O C I T Y B I A S

As shown above, with plane-wave modelling we can estimate the bias, that is, the relative difference between (fundamental mode) surfacewave phase velocities measured from EGF and GF for a known ambient noise energy distribution EP (ω, θ ) and medium. We illustrate thiswith an isotropic and a realistic azimuth dependent ambient noise energy distribution and with simple models for structural heterogeneityand anisotropy. The geometry of the experiments is given in Fig. 1, with ϕ the azimuth from A to B measured from north. Unless otherwisementioned, the period of the plane wave T = 30 s, the distance between the stations lAB = 480 km with central point fixed at O, and thebackground phase velocity c0 = 4 km s–1. In all examples we consider—for eq. (1)—incoming plane waves with θ from 0–360◦ in intervalsdθ = 0.5◦.

3.1 Homogeneous and isotropic medium

3.1.1 Isotropic energy distribution

For an isotropic distribution of ambient noise energy, that is, ∂EP (ω, θ )/∂θ = 0, and station pair AB in S-N direction (that is, ϕ = 0◦),Fig. 3(a) shows cross-correlation functions for plane waves arriving from 0◦ to 360◦. The maximum plane wave traveltime difference betweenA and B is δtmax = 120 s for a northward propagating plane wave (θ = 0◦ or 360◦), and δtmin = −120 s for a southward propagating planewave (θ = 180◦). For waves perpendicular to A–B (θ = 90◦ or 270◦), δt = 0 s. Following (1), the sum of individual cross-correlations givesthe cross-correlation function CAB(ω, t) for stations A and B (blue line, Fig. 3b). The time derivative of CAB(ω, t) then yields the EGF (reddashed line, Fig. 3b).

The appearance of two arrivals in CAB(ω, t) (and EGF) can be understood both with stationary phase (e.g. Snieder 2004) and Fresnelzone analysis. The individual cross-correlation functions from plane waves propagating with the azimuth near the two-station line, that is,small |θ − ϕ|, interfere (stack) constructively (near δt = ±120 s), whereas waves with larger |θ − ϕ| interfere destructively and do notcontribute to CAB(ω, t). With λ the wavelength, constructive interference occurs if

|AB cos(θ − ϕ) − AB| < λ/2, (10)which defines the first Fresnel for a homogeneous and isotropic medium, or

|φ(θ )| =∣∣∣∣{∫ B

B′

ω

c(ω, x)dl(θ ) −

∫ AA′

ω

c(ω, x)dl(θ )

}−

∫ BA

ω

c(ω, x)dl(ϕ)

∣∣∣∣ < π (11)for a general heterogeneous medium. In (11), φ(θ ) is the phase difference between the individual cross-correlation function from theincoming plane wave with azimuth θ and ray path l(θ ) and the individual cross-correlation function from the plane wave propagating alongthe interstation ray path l(ϕ) from A to B. Plane waves within the first Fresnel zone, that is, with θ satisfying (10) or (11), contribute mostto the construction of the GF. Plane waves outside the first Fresnel zone either cancel out through destructive interference (as in the exampleshown) or give spurious arrivals if destructive interference is incomplete owing to, for instance, uneven distribution of ambient noise energy.For the parameters given, the first Fresnel zone consists of two parts: 0◦ ≤ θ < 28.9◦ and 331.1◦ ≤ θ < 360◦ (yellow boxes, Fig. 3a) forthe construction of GAB(ω, t) and |θ–180◦| < 28.9◦ (red box, Fig. 3a) for the construction of GBA(ω, t). Here, the EGF (red dashed line,Fig. 3b) is nearly identical to the real GF (black line), with zero phase shift for both the causal (positive-time) and acausal (negative-time)part, as expected from theory (e.g. Weaver & Lobkis 2004; Roux et al. 2005; Nakahara 2006).

3.1.2 Uneven energy distribution

For uneven distribution of plane-wave sources we consider two cases. In the first, ambient noise generated plane waves only propagate alongthe direction of the two-station path, that is, EP (ω, θ )>0 for θ = ϕ or θ = π + ϕ and EP (ω, θ ) = 0 elsewhere. The cross-correlation functionobtained (blue line, Fig. 3c) is the same as in traditional two-station analysis (e.g. Yao et al. 2006) but has a π /4 phase advance comparedto the GF (black lines, Figs 3b and c). Since the EGF (red dashed lines, Fig. 3c) is π /2 phase delayed (away from zero time) with respect toCAB(ω, t), as expected from the negative time derivative, there is a π /4 phase shift between EGF and GF.



Figure 3. Illustration of the construction of Green’s function from 30 s period plane wave interferometry in a homogeneous and isotropic medium with phasevelocity c = 4 km s–1 at period 30 s. (a) Cross-correlagram for a south-north station pair apart from 480 km with isotropic plane wave energy distribution.Each horizontal line in the cross-correlagram gives the individual cross-correlation function, given by the integrand of eq. (1). The white colour represents thepositive values while the black colour shows for negative values. The red box shows the first Fresnel zone of plane wave interferometry, given by eq. (10),for the construction of the anticausal part Green’s function, while the yellow boxes shows that for the causal part Green’s function. (b) Comparison of thecross-correlation function (CF), empirical Green’s function (EGF), and the theoretical Green’s function (GF) for isotropic noise energy distribution with cross-correlagram shown in (a). CF (blue trace) is the stack of all individual cross-correlation functions, that is, vertical directional stack of the cross-correlagram in(a). EGF (red dashed trace) has π /2 phase shift to CF but is almost identical to the theoretical GF (black trace). (c) Comparison of the CF, EGF and GF whenplane wave energy is 1 only at azimuth angle θ = 0◦ and 180◦ but zero elsewhere. In this case, the EGF has π /4 phase shift away from zero time compared tothe GF.

For the second case we consider a more realistic, azimuth-dependent (normalized) ambient noise energy distribution (black curve,Fig. 4a), inferred from our study in SW China (Yao et al. 2009). The azimuth dependence of the (normalized) amplitude of the cross-correlation functions, CAB(ω, t), blue curve in Fig. 4(a), differs from that of the ambient noise energy (black curve), indicating that CAB(ω, t)cannot be used as a proxy for EP (ω, θ ). As a consequence, the phase velocity bias between EGF and GF (blue line, Fig. 4b) dependson azimuth: that is, μ = μ(θ ). The largest phase velocity biases occur where EP (ω, θ ) changes most rapidly with azimuth (for instance,between 0◦ and 90◦ and between 180◦ and 270◦, Fig. 4a), with up to 3 per cent bias for some station pairs (e.g. ϕ ∼ 20◦ or 70◦). For relativelysmooth azimuthal variation of ambient noise energy (e.g. 90◦–160◦ or 270◦–315◦) the phase velocity bias is small (


Figure 4. Bias of phase velocity measurements from plane wave interferometry in case of non-isotropic plane wave energy distribution in homogeneousmedium with or without azimuthal anisotropy: (a) azimuthal distribution of ambient noise energy EP (shown as the black curve) and normalized surface waveamplitudes ACF of the cross-correlation functions (blue or red curve); and (b) relative phase velocity bias (μ) between the EGF and GF as given by eq. (9). Theblue curve represents for the bias for a homogeneous and isotropic medium and the red curve for a homogeneous medium with 5 per cent azimuthal anisotropywith fast direction ψ = 45◦. The geometry of the station pair is described in Section 3.

Figure 5. Relative phase velocity bias (μ) [black line in (c)] for an isotropic ambient noise energy distribution [black line in (a)] in a homogeneous mediumwith 5 per cent azimuthal anisotropy with fast direction ψ = 45◦. In (a) the red line shows the normalized surface wave amplitudes ACF of the cross-correlationfunctions. In (b) the black line shows the interstation phase velocity with different azimuth angles.

evaluated with a ray theoretical approximation. In heterogeneous media the full wave GF may be different, but the (scale dependent) differenceis expected to be small.


Under the plane wave approximation, azimuthal anisotropy can cause phase shifts between EGF and GF, even for an isotropic energydistribution. The effect is small, however. For a homogeneous, 5 per cent azimuthally anisotropic medium (with fast axis at ψ = 45◦, alongthe NE–SW direction) the phase velocity bias μ(θ ) is generally less than 0.5 per cent (Fig. 5c). The bias vanishes for station pairs in thedirection of zero phase velocity gradient (in the first Fresnel zone), for example, ϕ = 45◦, 135◦, 225◦ and 315◦ (Figs 5b and c) and is largestfor the station pairs with the largest velocity gradient with respect to azimuth, for example, ϕ = 0◦, 90◦, 180◦ and 270◦ (Figs 5b and c).


To assess the effect of medium anisotropy on GF reconstruction and phase velocity bias in the presence of uneven energy distribution we usethe same EP (ω, θ ) as in Section 3.1.2 (black line, Fig. 4a). The variations with azimuth of (normalized) amplitude of the cross-correlationsand bias μ (red lines in Figs 4a and b) are virtually the same as in the isotropic case (blue lines, Figs 4a and b), with large biases occurringnear azimuths where ambient noise energy changes rapidly with respect to station pair azimuth ϕ. From tests like this we conclude that the



Figure 6. Geometry of plane wave interferometry for a heterogeneous and isotropic medium (Section 3.3.1) with isotropic ambient noise energy distribution.The velocity distribution is c(x, y) = c0 + c · exp{−0.5(x2 + y2)/r2}, with the background velocity c0 = 4 km s–1 shown as the white region and the 2-DGaussian shape anomaly centred at O(0, 0) shown as the black-shaded region with the characteristic horizontal scale r. The largest anomaly, c = 0.5 kms–1 for case 1 or c = −0.5 km s–1 for case 2, is shown with black. The two stations at A (0, −240) and B (0, 240) are shown as the black triangles. For ananomaly with the horizontal scale r = 50 km and c = 0.5 km s–1, the half width of the first Fresnel zone for interferometry (see eq. (11)) with respect to thetwo-station path (from A to B) is about 17◦, 29◦ and 38◦ for T = 10, 30 and 50 s, as bounded with the green, blue and yellow lines, respectively.

bias in phase velocity (from EGFs) due to (weak) azimuthal anisotropy is much smaller than that of uneven distribution of ambient noiseenergy.

3.3 Heterogeneous and anisotropic medium

Finally, we examine the effect of ambient noise energy distribution on phase velocity bias μ(θ ) in the presence of both azimuthal anisotropyand (2-D) medium heterogeneity.


Much like azimuthal anisotropy (previous section), medium heterogeneity can bias phase velocities measured from EGFs even for an isotropicenergy distribution due to the assumption of distant (plane-wave) sources. In a first experiment we consider a Gaussian anomaly—centred onO between A and B (Fig. 6)—expressed as c(x,y) = c0 + c exp{−0.5(x2 + y2)/r2}, where c0 = 4.0 km s–1 is the background wave speed,c (0.5 km s–1 for case 1 and −0.5 km s–1 for case 2) the magnitude of the anomaly, x and y spatial coordinates and r the characteristiclength (in km) of the anomaly. Since the velocity distribution is rotationally symmetric the average phase velocity for any station pair azimuthϕ will be equal; for the following calculation we take the station pair with ϕ = 0◦ (Fig. 6).

To investigate the dependence of bias on spatial scale (of medium heterogeneity), we fix the period of incoming plane waves T at 30 sand let r increase from 2 to 150 km (Fig. 7a). For case 1, that is c = 0.5 km s–1, the resulting phase traveltime difference δtAB between EGFand the theoretical GF is close to zero for r less than several kilometres or larger than ∼100 km, but δtAB will be as much as 1 s for r around35 km. The latter would correspond to a phase velocity measured from EGF that is ∼1 per cent slower than from GF. In case 1, δtAB ≥ 0 forall r, which implies that phase velocities estimated from EGFs are always less than the theoretical phase velocities. Likewise, for case 2(c =−0.5 km s–1), δtAB ≤ 0 for any r, indicating that the phase velocity will be overestimated.

This can, again, be understood with a simple Fresnel zone analysis. For the geometry given (Fig. 6), the average velocity will (forcase 1) be largest between the two stations. Plane waves propagating off the two-station path but within the first Fresnel zone (and thusrelevant for the construction of the EGF) will sense lower average velocities and thus render a larger travel difference δtAB than in ahomogeneous, isotropic medium, with phase velocity the average interstation velocity. This results in a lower phase velocity, that is, a phasevelocity bias μ < 0. Similarly, c < 0 leads to μ > 0. For small r the average phase velocity between A and B is close to c0 and for very largescale anomalies (here, r > 100 km) plane waves within the first Fresnel zone all propagate with speeds similar to the interstation average. Forboth cases, δtAB (within the first Fresnel zone) will be very similar to that in a homogeneous and isotropic medium, and therefore the finalphase velocity estimation from the EGF will be very close to that from the theoretical GF, that is, μ ∼ 0.



Figure 7. The phase traveltime difference δtAB between the EGF and the theoretical GF for a heterogeneous and isotropic model (Fig. 6) with isotropic ambientnoise energy distribution. In (a), we show δtAB as a function of the horizontal scale of the anomaly (r) with the dashed line for a positive anomaly (c =0.5 km s–1, case 1 in Fig. 6) and solid line for a negative anomaly (c = −0.5 km s–1, case 2 in Fig. 6). In (b), we fix the horizontal scale of the anomaly r =50 km and change the period of incoming plane waves. The dashed, solid, and dash-dotted lines show the results for a positive anomaly (c = 0.5 km s–1), noanomaly (c = 0) and a negative anomaly (c = −0.5 km s–1), respectively.

To investigate the dependence on spectral scale (in the data), we fix r at 50 km and let period T increase from 5 to 70 s (Fig. 7b). We setc0 at 4 km s–1 for all periods. For homogeneous and isotropic media (that is c = 0) δtAB is close to zero from T = 5 to 55 s (as expected);the negative values for T > 55 s reflect the breakdown of far field approximation for surface wave propagation (Yao et al. 2006) and, thus, theaccuracy of (6). For c > 0 (case 1) δtAB is positive (and μ < 0) and increases as T increases (from 5 to 55 s). For c < 0 (case 2) δtAB isnegative (and μ > 0) and the magnitudes of the traveltime and phase velocity biases (|δtAB| and |μ|, respectively) increase with increasing T .

Since the velocity distribution c(x,y) is the same for each period T , the width of the first Fresnel zone increases with increasing T (orwavelength λ) according to (11). The half width of the first Fresnel zone with respect to the two-station path is about 17◦, 29◦ and 38◦

for T = 10, 30 and 50 s, respectively (Fig. 6.) Narrower widths of the first Fresnel zone generally produce smaller bias δtAB and μ betweenstations A and B, whereas, longer period (longer wavelength) data produces larger differences between EGF and the theoretical GF, whichconfirms results by Tsai (2009).

In a second experiment we investigate a heterogeneous model (case 1 in Fig. 8) with positive and negative anomalies centred around(0, −100) and (100, 50), respectively. The geometry of the station pair AB is as before, and the ambient noise energy is initially azimuthallyisotropic (black line, Fig. 9a). For this model, the relative phase velocity bias at T = 30 s reaches −2 per cent for ϕ = 0◦ and 1 per centfor station pairs with azimuth angle ϕ ∼ 50◦ (Fig. 9c). In general, phase velocities are underestimated for station pairs with high averageinterstation phase velocities, whereas phase velocities from EGF are overestimated for low velocities (Figs 9b and c). In summary, for isotropicenergy distribution, and with asymptotic theory, the strength of phase velocity anomalies (with respect to a constant background) obtainedfrom EGFs is usually underestimated.


The velocity bias μ(θ ) produced by uneven energy distribution (black line, Figs 4a and 10a) in the presence of 2-D heterogeneity with andwithout 5 per cent azimuthal anisotropy (case 1 and 2 in Fig. 8) is presented in Fig. 10(b) (blue line for isotropic medium; red line forthe medium with 5 per cent anisotropy). For comparison, the green dotted curve shows the velocity bias for the heterogeneous medium butisotropic plane-wave distribution (Section 3.3.1) and the brown curve shows the bias for the homogeneous medium but uneven plane-waveenergy distribution (black line, Fig. 10a). The velocity bias for the heterogeneous (isotropic or anisotropic) medium with uneven plane-wave



Figure 8. Heterogeneous model (with or without azimuthal anisotropy) for investigating the Green’s function retrieval. The velocity model is c(x, y) =c0(x, y){1 + Ac cos 2(θ − ψ)}. The colour bar shows the value of the isotropic part of the velocity c0(x, y)(km s–1). The short bars, all with azimuth angleψ = 135◦ showing the fast propagation direction, give the azimuthally anisotropic part of the velocity with amplitude Ac = 0 for case 1 (no azimuthalanisotropy) and Ac = 5 per cent for case 2 (with azimuthal anisotropy).

Figure 9. Same as in Fig. 5 with isotropic ambient noise energy distribution in the heterogeneous medium without azimuthal anisotropy (case 1 of the modelin Fig. 8).

energy distribution consists of two parts (which are not entirely independent of each other): bias for homogeneous medium with unevenplane-wave energy distribution (brown dashed line, Fig. 10b) and bias for heterogeneous (isotropic or anisotropic) medium with isotropicplane-wave energy distribution (green dashed line, Fig. 10b). Our tests suggest, as before, that the former is generally (much) larger than thelatter.

4 R E C OV E RY O F A M B I E N T N O I S E E N E RG Y

4.1 Methodology

In the previous section, we have shown that uneven distribution of ambient noise (plane wave) energy can produce a substantial bias in phasevelocities measured from ambient noise interferometry (that is, from EGFs). So far, we have assumed a particular noise energy distribution(e.g. black curve in Figs 4a and 10a). In general, however, we do not know EP (ω, θ ) a priori. But we do know the interstation cross-correlationfunctions CAB(ω, t), or the EGFs inferred from them. The relationship between the amplitudes of CAB(ω, t) and EP (ω, θ ) is non-linear—asshown in (1) and, for instance, Fig. 4(a)-–and the former cannot be used as a proxy for the latter. We will show, however, that EP(ω,θ ) can be



Figure 10. Same as in Fig. 4 with uneven ambient noise energy distribution in the heterogeneous medium as shown in Fig. 8 with blue curves showing theresults without azimuthal anisotropy (case 1 in Fig. 8) and red curves showing the results with 5 per cent azimuthal anisotropy (case 2 in Fig. 8). In (b) thegreen dashed line shows the relative phase velocity bias with isotropic ambient noise energy distribution in the heterogeneous medium (case 1 in Fig. 8), sameas the black line in Fig. 9(c). The brown dashed line shows the relative phase velocity bias with uneven ambient noise energy distribution [black curve in (a)]in the homogeneous and isotropic medium with velocity 4 km s–1.

estimated from CAB(ω, t) and that the bias in phase velocity measurements from ambient noise interferometry can be quantified and reduced.This involves an inversion, which is here formulated in the context of the plane-wave modelling described in Section 2.

After discretization, eq. (1) in the main surface wave window can be rewritten as

Cn(ω, t, ϕn)Wn(t,n) =M∑

m=1Em(θm) cos [ω(t − δtnm)] Hnm(t, δtnm)Wn(t,n), (12)

where Cn(ω, t, ϕn) is the cross-correlation function for the nth (n = 1, 2, . . . , N) two-station pair with azimuth angle ϕn and interstationdistance n , Wn(t, n) the surface wave window function for the nth two-station pair (eq. 7), Em(θm) the plane wave energy at the mth azimuthθm (m = 1, 2, . . . , M) and δtnm and Hnm(t, δtnm) the phase traveltime difference and cosine taper window function of the nth two-station pairfor the mth incoming plane wave, respectively. Taking the Fourier transform for both sides of (12) gives

F{Cn(ω, t, ϕn)Wn(t, n)} =M∑

m=1Em(θm)F{cos [ω(t − δtnm)] Hnm(t, δtnm)Wn(t,n)}. (13)

Separation of the real and imagery parts (at frequency ω) gives the following matrix equation:

AR + jAS = (R + jS)E, (14)

where j is the imaginary unit, AR and AS N × 1 dimensional vectors, R and S N × M dimension matrices, and E an M × 1 dimension vector(AR)n = Re

(F{Cn(ω, t, ϕn)Wn(t, n)}

)∣∣ω

(AS)n = Im(F{Cn(ω, t, ϕn)Wn(t, n)}

)∣∣ω

Rnm = Re(F{cos [ω(t − δtnm)] Hnm(t, δtnm)Wn(t,n)}

)∣∣ω

Snm = Im(F{cos [ω(t − δtnm)] Hnm(t, δtnm)Wn(t,n)}

)∣∣ω

Em = Em(θm), (15)with Re(· · ·)|ωand Im(· · ·)|ω operators that take the real and imaginary part at frequency ω.

We note that eq. (14) is linear between data (AR and AS) and model parameters E because of the linear property of the Fourier transform.Therefore, E can be solved from AR and AS through a least-squares inversion scheme. However, the energy or amplitude of the windowedcross-correlation function is non-linearly dependent on E as inferred from (13) and (14). To obtain, by inversion, the ambient noise energyvector E we define the penalty function χ (E)

χ (E) = (RE − AR)T (RE − AR) + (SE − AS)T (SE − AS) + λD(DE)T (DE), (16)

where the superscript T denotes the transpose of a matrix, λD a damping parameter, and D an (M – 1) × M dimension smoothing operator(first finite difference matrix) defined as



D =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−1 1 0 ... ... 00 −1 1 ... ... 0

... ...

... ...

0 ... ... −1 1 00 ... ... 0 −1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

On the right-hand side of (16) the first and second terms give the data misfit for the real and imaginary parts as shown in (14), respectively,and the third term denotes model roughness. Optimization of this equation, that is, solving for ∂χ (E)/∂E = 0, gives the solutionE = (RT R + ST S + λDDT D)−1 (RT AR + ST AS) . (17)

Eq. (17) shows how the distribution of ambient noise energy can be estimated from the real and imaginary parts of the Fourier transformedcross-correlation functions.

4.2 Proof of concept

To illustrate the inversion for ambient noise energy E we use the same station geometry as before, with station azimuth ϕ (from A to B)varying from 0◦ to 358◦ with dϕ = 2◦ so that the number of data (cross-correlations) N = 180. For the discretization in (12) we use dθ= 0.5◦ to ensure enough sampling of plane waves for the GF recovery. Inverting for E at dθ = 0.5◦ intervals would imply 720 unknowns(M � N). In order to reduce the number of unknowns (to K = 90), however, we invert for Ẽ at 4◦ interval (that is, θ̃k = 0◦, 4◦, . . . , 356◦). Forany incoming plane wave with azimuth θm(0 ≤ θm < 360◦), there exists a number k such that θ̃k ≤ θm < θ̃k+1 and the energy Em is linearlyinterpolated between (Ẽ)k and (Ẽ)k+1 as

Em = βmk(Ẽ

)k+ βm(k+1)(Ẽ)k+1

βm(k+1) = (θm − θ̃k)/(θ̃k+1 − θ̃k)βmk = (1 − βm(k+1)), (18)or in a matrix form as

E = βẼ, (19)with β an M × K matrix with elements βmk ; if θm ≥ θ̃K = 356◦, θ̃K+1 = 360◦ and (Ẽ)K+1 =

(Ẽ

)1. Substituting (18) into (14) and reformatting

it, we obtain AR + jAS = (R̃ + j S̃)Ẽ, with R̃ and S̃ N × K dimensional matrices defined asR̃ = RβS̃ = Sβ. (20)

Therefore, the solution sought is

Ẽ = (R̃T R̃ + S̃T S̃ + λDDT D)−1 (R̃T AR + S̃T AS) . (21)We test this inversion scheme with the same noise energy distribution as before (black lines in Fig. 4a). We calculate the N = 180

cross-correlation functions using (1) and take the real and imaginary parts, that is,AR and AS , as in (15). With δtnm = δφnm/ω, where δφnmis the phase delay in (4), we obtain R̃ and S̃ from (15) and (20). Finally, we obtain estimates of the azimuthal distribution of ambient noiseenergy Ẽ from (21) for different damping parameters λD .

For a homogeneous and isotropic medium (as in Section 3.1) we can recover the ambient noise energy (magenta circles, Fig. 11a)without damping, that is, λD = 0. For a heterogeneous, isotropic medium (Fig. 8, case 1) the undamped solution (magenta circles, Fig. 11b)is unstable, but Ẽ is identical to distribution E for λD = 100 (blue circles, Fig. 11b). If the medium is azimuthally anisotropic (we added5 per cent anisotropy, as in Fig. 8, case 2) E can still be recovered but stronger damping is needed to stabilize the solution (Fig. 11c).

The above examples demonstrate that ambient noise energy EP (ω, θ ) can be recovered from data, that is the cross-correlations. Wehave, however, used the same model in the forward problem as in the inversion, whereas in practice the input model, for example, phasevelocity maps (Yao et al. 2006), will be an estimate of the true model. We recall that measurements (EGFs) are influenced by a combinationof uneven ambient noise energy distribution and medium complexity (heterogeneity and anisotropy). The influence from medium complexityis a consequence of our assumption of plane-wave sources, and is generally smaller than that from noise energy distribution (Section 3).To assess the recovery of EP (ω, θ ) when the input model for inversion is different from (but close to) the true model we calculate cross-correlation functions from one model (Fig. 8, case 2) and perform the inversion with another (Fig. 12a). The fact that recovery (blue circles,Fig. 12b) is satisfactory motivates the following three-step approach. First, we use the phase velocity maps obtained by inversion of theoriginal (uncorrected) data from the recovered EGFs (e.g. Yao et al. 2006) to estimate the spatial distribution of ambient noise energy. Second,we use this estimate of the noise energy distribution to calculate (and remove) the bias in phase velocities. Third, we use the corrected phasevelocities for final unbiased inversion. (This loop can be repeated, but our simulations suggest that a single iteration is sufficient.)



Figure 11. Recovery of the azimuthal distribution of ambient noise energy (magenta or blue open circles) from the cross-correlation functions as described inSection 4 for (a) the homogeneous and isotropic medium (Section 3.1) without adding damping (λD = 0), (b) the heterogeneous medium without azimuthalanisotropy (case 1 in Fig. 8) with the damping values λD = 0 (for the magenta circles) and λD = 100 (for the blue circles) and (c) the heterogeneous mediumwith 5 per cent azimuthal anisotropy (case 2 in Fig. 8) with the damping values λD = 100 (for the magenta circles) and λD = 10000 (for the blue circles). Theinput ambient noise energy EP is shown as the black line. The red line shows the amplitude of the cross-correlation functions.

5 A P P L I C AT I O N I N S E T I B E T A N D A Z I M U T H A L A N I S O T RO P Y

We use one month of vertical component data from the seismograph array in SE Tibet (Fig. 13a) to illustrate the process of recovery ofambient noise energy, estimation of phase velocity bias, and inversion for velocity model using an iterative procedure. We show the differenceof the isotropic phase velocity map and azimuthal anisotropy before and after the correction of phase velocity bias.

5.1 Iterative procedure

We obtain CAB(ω, t) for all possible two-station pairs (Fig. 13c) by one-bit cross-correlation of vertical component data recorded at A and Bin the period band 10–30 s—see Yao et al. (2008) for details about data processing. To obtain more symmetric (reciprocal) EGFs from thecross-correlation functions CAB(ω, t) we use eq. (5) and stack, for each station pair, the causal and anticausal parts of the EGFs. We refer tothis stack as the symmetric component. From the resulting EGF we measure the phase velocities c̃AB using the method by Yao et al. (2006).This assumes perfect recovery of GF in a far field, but—as discussed in Section 3—the obtained dispersion data may be biased due to unevennoise distribution and structure complexity. We use the following iterative procedure to estimate the ambient noise energy distribution, phasevelocity bias and 2-D phase velocity distribution at certain period T .

Step 1: Invert for 2-D phase velocity map (isotropic or azimuthally anisotropic) using interstation phase velocity measurements c(k) (k =1, 2, . . . ; k = 1 for the original measurements c̃AB from symmetric component EGFs, and k > 1 for updated phase velocities in Step 3 afterbias correction in each iteration)

Step 2: Estimate ambient noise (plane-wave) energy distribution Ẽ(k) by means of plane-wave modelling (Section 4). As point of departurefor the modelling we use the 2-D phase velocity maps from Step 1, and we use both the causal and anticausal part of the cross-correlationfunctions to obtain the data AR and AS in (15), with the requirement that the interstation distance is at least 2 wavelengths.



Figure 12. (a) A modified velocity model for the modelling in the inversion using eq. (21). The original velocity model is shown as the case 2 of the model inFig. 8, which is used for the calculation of the cross-correlation functions from eq. (1). The amplitude of the velocity anomaly (isotropic part) in the modifiedmodel is 75 per cent of that in the original velocity model. And a maximum of 0.05 km s–1 random noise is added to the isotropic part of the modifiedvelocity model. For the azimuthal anisotropic part, we add a maximum of 25◦ random noise to the azimuth of the fast propagation direction and a maximum of2 per cent random noise to the amplitude of azimuthal anisotropy (originally 5 per cent). (b) Recovery of the ambient noise energy (blue open circles) whenusing the modified velocity model (a) in the inversion but the original velocity model for calculating the cross-correlation functions. In (b), the input ambientnoise energy is shown as the black curve and the red curve shows the amplitude of the cross-correlation functions.

Step 3: Estimate the relative phase velocity bias μ(k) between the EGF and theoretical GF defined in (9) and update the phase velocitymeasurements with

c(k+1) = c̃AB/(1 + μ(k)). (22)Since we use the symmetric component EGFs for dispersion analysis, the phase delay between the EGF and GF expressed as eq. (8) can bemodified as

δφAB(ω) = {[G̃AB(ω, t) + G̃BA(ω, t)]W (t,AB)} − [GAB(ω, t)W (t,AB)]. (23)Note that we still use the 2-D phase velocity map from Step 1 for plane-wave modelling and for calculating the EGF and theoretical GF.

Step 4: If c(k+1) or Ẽ(k) converges, stop iteration; otherwise go back to Step 1.

We emphasize that the obtained ambient noise energy distribution Ẽ(k) depends on the normalization involved in cross-correlation (herewe use one-bit cross-correlation). In view of the relationship between the cross-correlation function and ambient noise energy (eqs (1) and(13)), Ẽ(k)(θ ) measures the overall normalized energy of the incoming plane waves at azimuth θ in the entire period of cross-correlation,which, in our example, is one month. This linearity also implies that the overall noise energy in a longer time window can be obtained throughsummation of noise energy in shorter time windows.

The convergence depends on the quality of the first preliminary phase velocity map from ambient noise tomography. If it is close to thereal model, we may expect convergence after one or two iterations. During the iterative process we can simultaneously invert for azimuthalanisotropy and the isotropic part of phase velocity maps. In the examples shown below we consider isotropy for the first two iterations (i.e.k = 1, 2) and subsequently (k = 3, 4, . . .) invert for azimuthally anisotropic phase velocity at the period T = 25 s. The corresponding raypath coverage at T = 25 s is shown as Fig. 13(b).

5.2 Results

In Fig. 14(a), we show the estimated ambient noise (plane-wave) energy distribution Ẽ(k) for the first four iterations, which is similar to theazimuth-dependent normalized amplitudes of cross-correlation functions for all possible MIT array station pairs (see Fig. 14b). The maximum



Figure 13. (a) Location of the MIT array in SE Tibet, China. (b) Ray path coverage for interstation phase velocity measurements at T = 25 s from theEGFs retrieved from one month data (2004 January) of the MIT array. (c) All interstation cross-correlation functions in the period band 10–30 s from one-bitcross-correlation of one month MIT array data (2004 January). The maximum amplitude of each trace is normalized for display purposes.

noise energy with respect to the array appears between the azimuth angle 90◦–135◦, which is about to the ESE of the array, pointing towardsthe oceans. The first two iterations, based on the isotropic phase velocity maps, yield similar values of Ẽ(k) almost at all azimuth angles. Similarresults are obtained for the third and fourth iterations, which are based on azimuthally anisotropic phase velocity maps. The incorporation ofazimuthal anisotropy yields slight differences in Ẽ(k) at some azimuth angles. This result suggests that one iteration is enough to obtain stableestimation of ambient noise energy both for isotropic and azimuthally anisotropic media. During the inversion for Ẽ(k), an appropriate valueof the damping parameter λD need to be set. Using an automated scheme, we determine λD from the trade-off curve (upper left-hand insetfigure in Fig. 14a) between the data misfit [dmisfit = (R̃Ẽ − AR)T (R̃Ẽ − AR) + (S̃Ẽ − AS)T (S̃Ẽ − AS), modified from the first two terms of(16)] and the model roughness [mrough = (DẼ)T (DẼ), modified from the third term of (16)]. We try a broad range of values for λD from 0to ATRAR + ATS AS (open circles in the inset figure of Fig. 14a) to obtain the trade-off curve by interpolation. On the trade-off curve we findλ1 such that dmisfit(λ1) = 0.15 max(dmisfit) and λ2 such that mrough(λ2) = 0.15 max(mrough). The value of λD for the final inversion of Ẽ is set,empirically, to λD = 100.5(log10 λ1+log10 λ2). The λD selected this way yields a smooth Ẽ and also a small data misfit from our tests.

Fig. 15 shows the relative interstation phase velocity bias μ(k) from EGFs for two periods T = 10 s (Figs 15a–d) and T = 25 s(Figs 15e–h) as a function of interstation distance (left-hand column) or azimuth (right-hand column) for the isotropic or azimuthallyanisotropic case. As expected, the bias μ(k) depends on azimuth (Figs 15b, d, f and h). For example, the systematic bias μ(k) for the paths withazimuth angles 90◦–135◦ (Figs 15f and h) is mainly due to the large variation of ambient noise energy between 90◦ and 135◦ (Fig. 14). Formost paths the bias is less than 1 per cent, probably because we use the symmetric component EGFs, which creates a more isotropic noisedistribution. The magnitude of the bias decreases with increasing interstation distance (Figs 15a, c, e and g). This is mainly because that thewidth of the first Fresnel zone for interferometry decreases as the interstation distance increases. As a consequence, the final construction ofGF will be less sensitive to the (relatively long wavelength) variation of ambient noise energy and medium heterogeneity. This effect is similarto the decrease in phase velocity bias due to a decrease in wavelength (and, thus, Fresnel zone width) discussed in Section. 3.3.1. Indeed, thebias at T = 10 s (Figs 15a and c) is smaller than at T = 25 s (Figs 15e and g). Consistent with the outcome of the modelling experiments(Section 3), the incorporation of (weak) azimuthal anisotropy has only a small effect on the estimation of bias for most paths (e.g. comparingresults between Figs 15b and d, or Figs 15f and h).



Figure 14. (a) Inversion results of azimuth-dependent ambient noise energy distribution at T = 25 s with respect to the MIT array using plane-wave modellingand an iterative approach. The results for the first four iterations are shown as the red, blue, green dashed and black dashed lines, respectively. The first twoiterations are based on the isotropic phase velocity maps and the third and fourth iterations use the azimuthally anisotropic phase velocity maps. In the upperleft-hand inset figure in (a), we show the trade-off curve between the data misfit and model roughness (defined in Section 5.2). The open circles show the trialdamping values in the inversion and the black solid square shows the final choice of damping value for the inversion of ambient noise energy. (b) Azimuthaldependence of the normalized amplitudes of the cross-correlation functions (bandpass filtered between 23 and 27 s) for all possible MIT array station pairs.The pie charts are constructed using the procedure from Stehly et al. (2006) by averaging the amplitude of all cross-correlation functions in each azimuthalsector (5◦ width here) with a geometrical spreading amplitude correction considering the difference in interstation distance. The colour bar in the right-handside gives the value of normalized amplitude. Note that the inversion results of ambient noise energy distribution (a) are similar to the azimuthal distributionof normalized amplitudes of the cross-correlation functions. The maximum noise energy appears between the azimuth 90◦–135◦, which is about to the ESE ofthe array and points towards the oceans.

Fig. 16 shows the phase velocity maps at T = 25 s obtained from the original, uncorrected phase velocity measurements c̃AB andfrom the corrected measurements c(k). Using c̃AB, we first perform inversions for isotropic and anisotroic phase velocity maps (Figs 16a andb) using the regionalization due to Montagner (1986). For the anisotropy inversion, the phase velocity at each grid point is expressed asc(x, y) = c0(x, y){1 + A(x, y) cos(2ψ) + B(x, y) sin(2ψ)}, where c0(x, y) is the transversely isotropic part of the phase velocity, A(x, y) andB(x, y) are the azimuthally anisotropic terms, and ψ is the azimuth of the ray path through (x, y). The isotropic part of the inversion thatalso allows for azimuthal anisotropy (Fig. 16b) is very similar to the map from the inversion that considers only isotropy (Fig. 16a), whichimplies that the trade-off between isotropic phase velocities c0(x, y) and azimuthally anisotropic terms A(x, y) and B(x, y) is small (see alsoSimons et al. 2002). In Fig. 16(c), we show the azimuthally anisotropic phase velocity map using the corrected interstation phase velocitiesc(k) after four iterations as given above, which is generally similar to the results using the uncorrected measurements c̃AB (Fig. 16b). Thelargest difference in isotropic phase velocity before and after bias correction is about 1 per cent, which is much smaller than the variation ofphase velocity map at this period (>10 per cent). The pattern and magnitude of the azimuthal anisotropy are also similar and the differenceis probably less than the uncertainty in the inversion results. Inversions for other periods yield similar results. In particular, the pattern andmagnitude of azimuthal anisotropy at 10 s are almost the same before and after bias correction.



Figure 15. Relative bias (μ) of phase velocities between the EGF and GF versus interstation distance (left-hand column) or azimuth (right-hand column) forthe periods T = 10 s (a–d) and T = 25 s (e–h) using either isotropic phase velocity maps (a, b, e, f ) or azimuthally anisotropic phase velocity maps (c, d, g, h)for the forward (plane-wave) modelling.

6 D I S C U S S I O N

Ambient noise tomography has become an important tool for investigations of the structure of the crust and shallow mantle lithosphere usingdata from seismograph array stations. However, the surface wave GFs are generally not fully reconstructed due to uneven distribution ofambient noise sources. This raises the concern that dispersion measurements from such EGFs are biased, which can degrade the accuracy ofambient noise tomography, especially for anisotropic structure.

We have investigated this problem under the assumption of plane wave propagation (that is, asymptotic theory). In particular, thetheoretical GFs used here are evaluated with a ray theoretical approximation. The approach presented here is generally applicable for ambientnoise array tomography if: (1) the waves generated from distant noise sources can be approximated as plane waves propagating across thearray (that is, the horizontal scale of the study region must be relatively small compared to the distance to the main noise sources—forexample, ocean microseisms—and the energy of local scattered surface waves must be much weaker than that of faraway sources); (2) thereis good azimuth coverage of two-station paths and (3) an adequate initial model for plane wave modelling is available.

The small-scale array in the first requirement is generally satisfied for most regional ambient noise array tomographic studies, forexample, in SE Tibet (Yao et al. 2006, 2008), New Zealand (Lin et al. 2007), South Korea (Kang & Shin 2006), and southern California(Shapiro et al. 2005). For ambient noise tomography, the period range is typically within 10–40 s, for which the ambient noise is generatedmainly through ocean wave activities (e.g. storms) generated in the northern oceans during the northern hemisphere winter and in the southernoceans during the northern summer (e.g. Stehly et al. 2006; Rhie & Romanowicz 2006). It is still under debate whether the seismic noise forperiods larger than 10 s is related to ocean wave activity in deep water (Stehly et al. 2006) or generated by the non-linear interaction of oceanwaves with seafloor near coastlines (Yang & Ritzwoller 2008a). Such nearby noise sources may invalidate plane wave modelling, especiallyat short periods (


Figure 16. Distribution of 2-D phase velocity maps (in per cent with respect to the average value) at T = 25 s using (a) uncorrected phase velocity measurementswithout azimuthal anisotropy inversion, (b) uncorrected phase velocity measurements with azimuthal anisotropy inversion, and (c) corrected phase velocitymeasurements with azimuthal anisotropy inversion as described in Section 2. The background image of (d) shows the difference (in per cent) of phase velocitiesbetween the isotropic part of phase velocity maps of (b) and (c). On (d) the thick black bars and red thin bars show the comparison of azimuthal anisotropyfrom (b) and (c), respectively. The colour bar gives the magnitude of perturbation (in per cent) of isotropic phase velocities and the two horizontal bars at thebottom of (b–d) gives the magnitude of azimuthal anisotropy.

the array in SE Tibet correlates well with distant ocean wave activity with seasonal variations. Since the array in SE Tibet is at least about onethousand kilometres away from the coastline, even if there exist near coastal sources, plane wave approximation is still a good approximationto represent energy propagation through this small array. The requirement that local scattering is small compared to the energy from distantnoise sources is not easily verified. For the array in SE Tibet, however, the dominant sources (after one-bit normalization of data) are oceanmicroseisms, with much weaker contributions from local scattering (Yao et al. 2009).

The second and third requirements are needed to ensure robust estimation of the azimuthal distribution of ambient noise energy. Forseismic arrays with good spatial and azimuthal interstation path coverage, such as the MIT array in SE Tibet (Yao et al. 2008), we can usuallyestimate noise energy at azimuth intervals of several degree. However, for seismic arrays with a narrow azimuthal path coverage, such a 2-Darrays with a large aspect ratios, the estimation of azimuthal energy distribution will be less reliable. As input velocity model for the planewave modelling one can use the isotropic phase velocity maps from ambient noise tomography without bias correction.

We note that in heterogeneous media the true (full wave) GF may be slightly different, with the difference expected to be dependent onthe scale of the medium. The simplified approach presented here is relevant, however, because most current ambient noise (surface wave)tomography studies similarly rely on asymptotic theory (e.g. ray paths, phase velocity maps). There is no fundamental obstruction to extendthe same concept to full wave theory, with the use of full wave sensitivity kernels, but that would only be useful if the entire tomographicinversion is posed as a full wave, multiscale problem (with model parameters inferred directly from broad-band data and not from phase orgroup velocity maps).

Our analysis suggests that ambient noise tomography for isotropic wave speed variations is likely to be robust (that is, it is relativelyinsensitive to bias due to incomplete GF reconstruction) if one uses long time windows (e.g. several months or even 1 yr) for cross-correlationand the symmetric component EGF (the sum of the causal and anticausal part of EGF) for dispersion analysis (e.g. Yang et al. 2008;Yao et al. 2008). For the MIT array in SE Tibet the phase velocity bias is generally less than 1 per cent, which is small compared to theinferred wave speed variations and which causes a very small effect on the isotropic phase velocity maps in the wave period of interest



(T = 10–30 s). The azimuthal dependence of phase velocity bias is a bigger concern for inversions for (azimuthally) anisotropic structure.Our example in SE Tibet shows, however, that this effect is generally small, and the (spatial) smoothing employed in surface wave tomographyfurther suppresses any effects of azimuth-dependent bias.

Even if the effect is small for our array in SE Tibet, we recommend that inversions for azimuthal anisotropy are subjected to bias analysisif the EGFs indicate that the distribution of ambient noise energy varies rapidly with azimuth. The azimuth dependence can also be quantifiedby means of beam forming (Harmon et al. 2008; Yao et al. 2009). Similar in concept to other differential methods, if ambient noise sourcesgenerate both Love and Rayleigh waves, radial anisotropy inferred from the discrepancy between Love and Rayleigh wave GFs is less sensitiveto the actual distribution of ambient noise energy. If Love and Rayleigh waves are excited by different source distributions, however, carefulanalysis of phase velocity bias for both Love and Rayleigh dispersion is necessary to constrain radial anisotropy.

We note that our plane-wave modelling approach assumes a lossless medium; that is, it ignores the effect of attenuation on the amplitudeof waves. We emphasize, however, that even without attenuation there will be an apparent decay of amplitude of EGFs due to the fact that thewidth of the first Fresnel zone for constructive interference decreases as the increase of interstation distance (see also Harmon et al. 2007).Recent studies (Prieto & Beroza 2008; Prieto et al. 2009) show that it is possible to extract anelastic (attenuation) structure from ambientnoise interferometry, provided that the noise energy distribution is isotropic. Since the noise energy distribution is generally not isotropic, ourinversion approach for estimating the ambient noise energy distribution could help interferometric quantification of medium attenuation.

7 S U M M A RY

We have presented a method for estimating the distribution of ambient noise energy and the bias in phase velocities from ambient noiseinterferometry under the assumption of plane wave propagation. Through an iterative approach we correct the phase velocity measurementsfrom EGFs and estimate the azimuthal anisotropy of surface wave propagation with more confidence. Our method can be applied to small-scalearrays with good spatial and azimuthal path coverage and which are located far from the dominant noise sources. With real application toSE Tibet, we find the azimuthal variation of ambient noise energy has very small effect on the isotropic and azimuthally anisotropic phasevelocities in SE Tibet.

A C K N OW L E D G M E N T S

We thank the Editor Jeannot Trampert and an anonymous reviewer for their constructive comments, which helped us improve the manuscript.We also thank Professor Jean-Paul Montagner at IPGP for providing the code for inversion of surface wave azimuthal anisotropy and DrVictor Tsai at Harvard for helpful discussions. This work was supported by US AFRL grants FA8718–07-C-0001.

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